Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

B. Quasiprobability distributions using squeezed-state expansionsB.2 Squeezed-state P functionsWe define the squeezed-state P function P ξ via the expansion∫ˆρ = dαdα ∗ P ξ (α, α ∗ ) ∣ 〉〈 ∣ α, ξ α, ξ ∫= dβdβ ∗ P (β,β ∗ ) ∣ 〉〈 ∣ β β (B.8)for the system density operator ˆρ.Operator correspondences for â and â † can be found by expressing these operators interms of ˆb and ˆb † and then using the usual P function operator correspondences. Proceedingthus we arrive at:âˆρâ † ˆρˆρâ †ˆρâ⇐⇒⇐⇒⇐⇒⇐⇒(α + ν ∂ )P ξ∂β(α ∗ − µ ∂ )P ξ∂β(α ∗ + ν ∂ )∂β ∗ P ξ(α − µ ∂ )∂β ∗ P ξ .(B.9a)(B.9b)(B.9c)(B.9d)These reduce to the usual coherent-state P function correspondences when ν → 0andµ → 1. When we are not in this limit, then extra derivative terms appear in the correspondences,leading to diffusion terms as well as drift terms in the evolution equation forP ξ from the linear terms in the master equation, and higher-order derivatives from thenonlinear terms. For example, the evolution equation for an anharmonic oscillator (Eq.(7.26)) is∂P ξ∂t[= E 0 − ∂∂α α + ∂∂α ∗ α∗ − ∂2∂β 2 µν ++ κ]∂2∂β ∗2 µν∗ P ξ[−2 ∂∂α |α|2 α +2 ∂∂α ∗ |α|2 α ∗ + D ξ (α, α ∗ )]P ξ ,(B.10)where D ξ contains diffusions terms as well as third- and fourth-order derivate terms, whichmust be truncated before Langevin equations can be generated for α and α ∗ . As in theWigner technique, the final phase-space equations are then approximate.A more fundamental problem with this squeezed-state P function is that the squeezingparameter ξ is constant and not part of the evolving dynamics. For this representation tobe of any advantage, we would need to know apriorithe value of the squeezing parameter174